Problem Statement: Let be a sequence of continuous functions defined on and a real valued function defined on . (a) Prove if uniformly then and (b) Does this hold is pointwise? Prove or give a counter example.
(a) Proof: We are given that uniformly and so since each is continuous it follows that is continuous since our convergence is uniform. Thus, is Riemann Integrable on .
To prove the given statement we will prove an equivalent statement, that .
Let . Since uniformly there exists some such that for every and . Consider the following for :
Thus we have shown that and so we have proven the original statement.
(b) Solution: Claim, this does not hold if pointwise. Consider the following counterexample:
Let’s consider the series of functions the describe “moving triangles”. Let describe the triangle with base from and height 2 centered at . Then the area of this triangle, or it’s integral, is 1. Let describe a triangle with base from and height 4 centered at . Let be defined to be zero on . Then the area of this triangle, and thus it’s integral, is also 1. We may continue in such a way, constructing triangles each with area 1 but with smaller and smaller bases on the x-axis. So, pointwise but for every . Thus,
Reflection: The proof part of this was not the hardest part for me. I hope that’s a good sign 🙂 The tricky part was coming up with the counter example. Thank you to Richard for sharing this one with me. It’s one of those nifty things to have up your sleeve.
Some other notes/thoughts about the exam. I am so ready to take this thing! I got the stomach flu this past Monday (5 days before the exam…ugh) and I was so upset, but today is my first day back to fully functioning life and I am happy to say that my analysis knowledge survived the flu! Now I am just ready to get this thing over! If anyone reading this has any last minute comments or analysis ideas they think I need to know, please feel free to share them in a comment on this post! I would be much appreciated 🙂